In this article, the predefined-performance (PP) trajectory tracking control problem is investigated for a class of nonholonomic mobile manipulator systems subject to modeling errors, system uncertainties, and external disturbances. Extended from predefined-time stability, PP stability is developed for the first time. In the proposed PP control framework, a predefined-time segmented time-varying sliding mode (SM) without reaching time is constructed to ensure faster zero-error tracking at the predefined time. Then, a performance function related with the symbol of error is designed to assure that the system tracking errors converge to the predetermined boundary within the preset time without evident overshoot. In this way, the PP stability is realized. A significant feature of this article is that the PP stability is guaranteed theoretically and practically in the presence of system uncertainties without utilizing any approximation function. Finally, simulation and experiment comparisons among the proposed PP control and existing conventional finite/fixed-time control are conducted to demonstrate the superior performance of the proposed control scheme.